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I have a cartridge heater with the following specs: 50W and 220V AC.

I would like to control it with a rheostat, but I am not sure how to pick an appropriate rheostat. Could someone give me a hand?

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As others have pointed out using a rheostat to control the heater is not the best way to solve this problem due these factors:

  1. It wastes a lot of power.
  2. The component is costly.
  3. There is a question of safety.

There is a better way to deal with the control of the heater. Even better than the AC phase control that is typically used for controlling lights. This better method takes advantage of the fact that heaters generally have a long time constant and so fairly low frequency control is feasible.

I would suggest that you get a SSR (solid state relay) that has optical isolated input control (this helps to address the safety factor) and implements zero crossing switching (this helps eliminate the electrical noise created by AC phase control switching). Use a simple microcontroller board to manage the SSR input such that you plan the ON / OFF control across a certain number of half cycles of the AC mains. Lets say you shoot for control period of 0.4 seconds. At a 50Hz line frequency this translates to a total control period of 40 half cycles of the AC line. With suitable input drive signal to the SSR you can have 40 steps of power control.

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You could use a rheostat. Your heater is about 1k ohm, so a 0-1K series resistance would allow you to reduce the power to 12W minimum (and 12W would be dissipated in the rheostat). Here is a typical datasheet. Note the current specification, so you'd need a 75W pot to keep the current within spec. Such a rheostat/potentiometer will cost you in excess of $60 US most likely (at least from North American sources).

Chances are that you would be better off using a phase control such as an inexpensive incandescent light dimmer. Not only would it be much cheaper, but it would run very cool.

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What is the resistance of the heater (Rheat)? What is the minimum and maximum power you want to put out of the heater?

Start from here and solve for Rstat (Rheat is the heater resistance, P is desired power):

P = (V^2)/R = ((220 * Rheat/(Rheat + Rstat))^2)/Rheat

Now evaluate Rstat at the minimum and maximum power points and bam, there's your rheostat range. The rheostat should have a power rating based on

P = (v^2)/Rstat

Where you use the minimum rheostat resistance and give some extra for safety.

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First, you need to consider your control circuit. In this case, you'll do something like

schematic

simulate this circuit – Schematic created using CircuitLab

Now, what is your desired operating range? In this case, what is the lowest power setting you'll need? Let's assume 10% of full power. This will require a rheostat of approximately 2 times the nominal heater resistance. Using the formula for a voltage divider $$V =VLINE (\frac{RL}{RL+RH}) $$ when Rh is 2 times RL, the minimum voltage will be 1/(1+2), or 1/3 of regular voltage, and the power will be 1/9, or 11%. For RH equal to 3 times RL, the minimum voltage will be 1/(1+3) or 1/4, and the minimum power will be 1/16, or about 6%.

There are a few problems here. Most importantly, at half power the rheostat and the load will dissipate equal power, so for a nominal 50 watt heater you'll need a 12.5 watt rheostat. Since you should NEVER run components right at their assumed limits, you'll need a bigger rheostat, and the next standard size up is 25 watts. Have you sized one? Typically, that size will run about 1 1/2 inches in diameter and 1 1/2 inches deep. It must be mounted with good airflow available. Much worse - it will be directly connected to mains, which means that you'll need to be very careful about mounting and wiring.

So, yeah, you can do it. As Spehro points out, you'll spend on the order of 60 bucks for a new one. There are probably better choices.

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I wouldn't want to control it with a rheostat, this wastes a lot of energy, gets very hot and still costs a lot of money. Just use a ordinary dimmer, like the ones for light bulbs. It saves some energy and you don't have to calculate much. Just be sure that it can handle 50 Watts.

If however you really want to use a rheostat, you could calculate the needed power dissipation as @WhatRoughBeast pointed out.

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